Recently, many studies have been devoted to questions of regularity under Zhikov's
[1]} and Fan's [2]} logarithmic
condition, when \(\lambda (r)\leqslant L<+\infty \)  in (REF )–(REF )
(see e.g.
[3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}),
and now the elliptic theory has reached a completely satisfactory form
(see original surveys
[18]}, [19]}, [20]}, [21]}
and monograph [22]}).
In general, the logarithmic condition has significantly advanced the theory of function spaces
with variable exponents [23]}, [24]},
which are an integral part of the generalized
Orlicz (Musielak-Orlicz) spaces [25]}, [26]}.
Nonlinear differential equations in Muselak-Orlicz spaces is a fairly extensive topic
for research (see the informative survey [27]}).
Here we only cite the recent papers
[28]}, [29]}, [30]}, [31]}
on Hölder's regularity and Harnack's inequality under the
Musielak-Orlicz assumptions and logarithmic conditions in the elliptic framework.
